1. Proportionality

2. Direct Proportionality What is it? Direct proportionality is when one thing is directly proportional to another. Two quantities are in direct proportion if one quantity increases at the same rate as the other.

3. Y is proportional to X This is usually written as y x, or y=kx, where k is the constant or proportionality. Example A varies directly with b. Given that b is 10 when A is 35, find a formula for A in terms of b. Then use this formula to find the value of A when b is 13. The value of b when A is 70.

4. Answer…. To work out this question, the following calculation would be needed. A b A = 3.5 x b A = k x b = 3.5 x 13 35 = k x 10 = 45.5 k = 35 / 10 K = 3.5 70 = 3.5 x b b = 70 / 3.5 A = 3.5 b = 20

5. Direct proportion involving x² One quantity can also be proportional to the square of another quantity. Example Given the y varies directly with the square of x and that y is 75 when x is 5, find y when x is 15.

6. Answer… Y = kx² y = 3x² 75 = k x 5² y = 3 x 15² 75 = 25k y = 675 k = 3

7. Inverse Proportion What is it? When quantities are in inverse proportion, one of the quantities increase as the other decreases, and vice versa. ‘ y is inversely proportional to x ’can be written as y 1 / x, where 1 or k is the constant of proportionality. If two variables are in inverse proportion, the product of their values will stay the same.

8. Example T is inversely proportional to r when T is 5, r is 4. Form a relationship between T and r and use it to find T when r is 40

9. Answer… T 1 / r T = 20 / r So T = k / r T = 20 / 40 T = ½ 5 = k / 4 k = 20

10. Inverse proportion involving x² One quantity can be inversely proportional to the square of another. For example the number of tiles needed to cover a wall is related to the area of the tile. As the area of the tile gets larger, the number of tiles needed gets smaller. Also the area of the tile is proportional to the square of its length, l². So the number of tiles, N is inversely proportional l² : N 1 / l²

11. Example The force of attraction, F, between two magnets is inversely proportional to the square of the distance, d, between them. Two magnets are 0.5cm apart and the force of attraction between them is 50 newtons. When the magnets are 3 cm apart what will be the force of attraction between them?

12. Answer… F 1 / d² F = 200 / 3² So F = k / d² = 22.2 newtons 50 = k / 0.5² k = 50 x 0.5² k = 200